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Math Help - Geometry of Systems

  1. #1
    Len
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    Geometry of Systems

    Show that all vectors in the vector field F(y,v)=(v,-y) are tangent to circles centered at the origin.

    The hint I have is I can use slopes or dot product: I just need some help on how to use them.

    Thanks in advance.
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  2. #2
    Super Member
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    I'm gonna' give this a whirl alright. So the equation of a circle is:

    x^2+y^2=r^2

    differentiating:

    2x+2yy'=0

    or:

    \frac{dy}{dx}=-\frac{x}{y}

    or parametrically:

    \frac{\frac{dy}{dt}}{\frac{dx}{dt}}=-\frac{x}{y}

    and thus:

    \begin{aligned}<br />
\frac{dy}{dt} &=-x \\<br />
\frac{dx}{dt} &=y\end{aligned}<br />

    Also, you've got to get Mathematica or something else and just plot these to see globally what's happening. In Mathematica 7, you could use:

    StreamPlot[{-x,y},{x,-3,3},{y,-3,3}]

    ain't that simple or what?
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